We present efficient parallelisation strategies for geometric multigrid solvers on GPUs. Such solvers are a fundamental building block in the solution of PDE problems using discretisation techniques like finite elements, finite differences and finite volumes. Generalised tensor product meshes, unstructured meshes and their block-structured combination are considered. Special focus is placed on numerically strong smoothers, which are challenging to parallelise due to their inherently sequential, recursive character. However, many practical problems require strongsmoothers, in particular in the presence of anisotropies in the differential operator, the underlying mesh, or both. We address the inherent trade-off between numerical and hardware performance, i.e., between global coupling and degree of parallelism. To further improve performance, a mixed precision strategy is applied. By carefully balancing these contradictory requirements, we achieve more than an order of magnitude speedup over highly optimised CPU code for a number of challenging test problems, without affecting the numerical performance of our solver.